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Functional complexity emerging from anatomical constraints in the brain: the significance of network modularity and rich-clubs

Gorka Zamora-López, Yuhan Chen, Gustavo Deco, Morten L. Kringelbach, Changsong Zhou

arXiv:1602.07625v2q-bio.NC

TL;DR

The relationship between network complexity and the capacity to integrate and segregate information remains unresolved, including whether modularity and rich-clubs contribute to complexity. The paper measures functional complexity across real and synthetic networks and finds that both anatomical features accompany greater complexity, while human resting-state dynamics match the largest complexity their connectome can host.

  • Problem

    How network complexity relates to integration and segregation remains unresolved, including whether modularity and rich-clubs contribute to complex dynamics.

    Source: §I · p. 2

  • Method

    The authors measure functional complexity from pairwise interaction variability and compare real connectomes and synthetic models across coupling strengths and topology perturbations.

    Source: §V · p. 10

  • Results

    Real networks show reduced complexity when modularity or hubs are disrupted, and human resting-state dynamics match the largest complexity their anatomical connectome can host.

    Source: §V · p. 10

  • Takeaways & Limitations

    The findings support a connection between modular organization, rich-club hubs, and the complex dynamics available to neural connectomes.

    Source: §V · p. 10

    The conclusions are constrained by a heuristic exponential mapping that estimates time-averaged correlations, not temporal fluctuations, and treats nodes as passive relays.

    Source: Limitations · p. 11

Abstract

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The large-scale structural ingredients of the brain and neural connectomes have been identified in recent years. These are, similar to the features found in many other real networks: the arrangement of brain regions into modules and the presence of highly connected regions (hubs) forming rich-clubs. Here, we examine how modules and hubs shape the collective dynamics on networks and we find that both ingredients lead to the emergence of complex dynamics. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which either modules or hubs are destroyed, we find that functional complexity always decreases in the perturbed networks. A comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, lies in a dynamical state that reflects the largest complexity its anatomical connectome can host. Last, we generalise the topology of neural connectomes into a new hierarchical network model that successfully combines modular organisation with rich-club forming hubs. This is achieved by centralising the cross-modular connections through a preferential attachment rule. Our network model hosts more complex dynamics than other hierarchical models widely used as benchmarks.

I. INTRODUCTION

The paper addresses how modules and hubs relate to network complexity, noting that common hierarchical models capture modularity but lack hubs. It defines functional complexity from pairwise correlation distributions and examines when complex collective dynamics emerge.

I. INTRODUCTION

  • Existing recursive hierarchical models capture modules but lack hubs, whereas connectomes centralize cross-modular links through highly connected rich-club regions.Source: §I · p. 1
  • Prior optimization produced interconnected communities but no hubs or rich-clubs; modular-network dynamics are most complex at an intermediate balance of inter- and intra-module links.Too few cross-module links leave dynamics segregated, while too many promote global synchrony.Source: §I · pp. 1–2
  • The paper tests one direction of the proposed relationship: whether centralizing cross-module connections through rich-clubs enhances functional complexity.Source: §I · p. 2

II. MEASURING FUNCTIONAL COMPLEXITY

  • Complexity emerges at intermediate collective states, where correlations span a broad range rather than clustering near zero or one.Source: §II · p. 2, Figure 1 · p. 3
  • Functional complexity is the deviation of pairwise-correlation distributions from uniformity, capturing broad intermediate distributions rather than independence or global synchrony.The measure compares the observed distribution with a uniform distribution across bins; the diagonal entries are excluded.Source: §II · pp. 2–3
  • The chosen measure was more sensitive to topology and robust to bin-count variation than alternatives, and can also use other pairwise functional-connectivity metrics.Source: §II · p. 3

III. FUNCTIONAL COMPLEXITY OF NEURAL CONNECTOMES

The authors estimate functional connectivity from anatomical connectomes across coupling strengths and compare complexity in real networks and surrogates. Real connectomes show greater complexity than surrogate networks, while selective rich-club lesions reduce it.

III. FUNCTIONAL COMPLEXITY OF NEURAL CONNECTOMES

  • The study estimates functional connectivity from structural connectomes using an analytical exponential mapping while increasing coupling strength, g, to examine the transition from independence to synchrony.The mapping introduces exponential decay over longer paths to prevent divergence at strong coupling.Source: §III · pp. 3–4

A. Comparison to surrogate networks

  • Across the connectomes, real networks achieve greater complexity than random, degree-preserving rewired, and modularity-preserving surrogate networks.The figure summarizes peak complexities; absolute values are not comparable across species because connectome sizes and densities differ.Source: Figure 2 · p. 4, A. Comparison to surrogate networks · p. 4
  • Selective rich-club-link removal lowered complexity in all four connectomes; no random lesion did so in C. elegans or macaque, versus 1.5% for cat and 22% for humans.Random lesions removed the same number of links and excluded rich-club links.Source: Table I · p. 5, A. Comparison to surrogate networks · p. 5
  • Real networks transition to synchrony more slowly than surrogates, leaving a wider coupling range in which complexity remains high.Source: A. Comparison to surrogate networks · p. 5

B. Complexity of human resting-state FC

The human resting-state functional connectivity is consistent with peak complexity supported by its anatomical connectome, within cross-subject variability. Synthetic-network comparisons show that complexity depends on both degree structure and modular organisation, with modular networks having an optimal internal-to-external link ratio.

B. Complexity of human resting-state FC

IV. FUNCTIONAL COMPLEXITY OF SYNTHETIC NETWORK MODELS

  • The synthetic-network analysis estimates correlation matrices from network structure across coupling strengths, then calculates functional complexity from those matrices.Source: §IV · p. 6

B. Modular networks

A. Random and scale-free networks

  • Scale-free networks have higher peak complexity than random graphs, while peak complexity decreases with link density in both network types.In scale-free networks, hubs synchronize before the remaining nodes, creating an intermediate-coupling state in which synchronized hubs coexist with weakly correlated nodes.Source: Figure 4 · p. 6, A. Random and scale-free networks · p. 7

B. Modular networks

  • Increasing modularity raises the coupling needed for global synchrony and produces bimodal correlations, while complexity does not increase monotonically with modularity.Source: B. Modular networks · p. 7

C. Hierarchical and modular networks

The paper compares three hierarchical and modular network models with degree-preserving random and rewired benchmarks. Its Centralised HM model combines nested modules with hub-centralized inter-modular connections and achieves higher complexity than the other tested hierarchical models.

Fractally hierarchical and modular networks

Random hierarchical and modular (RHM) networks

Centralised hierarchical and modular (CHM) networks

  • The Centralised HM model replaces random inter-modular links with preferential attachment, assigning nodes external-link probabilities proportional to their within-module rank.The inter-modular links use exponents γ2 = 2.0 and γ1 = 1.7 to produce rich-clubs comparable to those in brain connectomes.Source: Centralised hierarchical and modular… · p. 8
  • C = 0.57, the Centralised HM model exceeds the other HM networks investigated and has larger complexity than standard hierarchical models.Its complexity decays more slowly at strong coupling than in random and rewired networks, indicating robustness against accidentally entering global synchrony.Source: Centralised hierarchical and modular… · pp. 8–10, Figure 7 · p. 9

V. SUMMARY AND DISCUSSION

The discussion links modular organization and rich-club hubs to functional complexity, while cautioning that their relationship to information integration and segregation is not fully established. It also compares human resting-state dynamics and a new hierarchical model with the complexity their connectomes or benchmark networks can support.

V. SUMMARY AND DISCUSSION

  • Functional complexity measures variability in functional interactions and emerges at intermediate states, where dynamics form interacting clusters rather than independence or global synchrony.It vanishes in both of those extremal cases.Source: §V · p. 10
  • Disrupting modular structure while conserving degrees, or vice versa, reduces functional complexity; the high-complexity regime is robust to shifts toward global synchrony.Selective removal of links between hubs also reduced complexity significantly more than random lesions.Source: §V · p. 10
  • Within the limits of the nonlinear mapping, human resting-state dynamics match the largest functional complexity their anatomical connectome can host.The origin and detailed role of resting-state dynamics remain debated.Source: §V · p. 10
  • The new hierarchical model combines nested modules with highly connected hubs and achieves higher functional complexity than benchmark models, including the Ravasz–Barabási model.The Ravasz–Barabási model’s hubs are disassortative, unlike the assortative rich-club hubs observed in brain connectomes; the new model centralizes cross-module connections through preferential attachment.Source: §V · p. 10

Complexity, modules, hubs, integration and segregation

  • The proposal that rich-club hubs integrate information across segregated modules remains a working hypothesis; topology alone does not explain the integration step3,20,32–34.The discussion points to information-propagation models as relevant to understanding integration36,37.Source: Complexity, modules, hubs, integration… · p. 11
  • The causal relationships between network complexity and the capacity to integrate and segregate information remain open questions, so the discussion cautions against treating them as established.The authors specifically advise against taking these relations for granted when interpreting clinical conditions39,40.Source: Complexity, modules, hubs, integration… · p. 11

Limitations

The limitations concern temporal aspects of functional complexity and the heuristic connectivity mapping, while the outlook addresses clinical use and future network models. The methods describe the connectome datasets and processing used for the analyses.

Limitations

  • The study measures interaction complexity, not the temporal evolution of functional complexity or the complexity of individual signal time-courses.The authors distinguish interaction complexity from temporal complexity, which may differ even for chaotic or periodic signals.Source: Limitations · p. 11
  • The heuristic exponential mapping estimates time-averaged cross-correlation and treats nodes as passive relays, limiting evaluation of temporal fluctuations and active nonlinear transformations.It incorporates some high-order contributions and avoids divergence without specifying a particular dynamical model, enabling computationally efficient comparisons.Source: Limitations · pp. 11–12

Outlook

  • The authors present functional complexity as a candidate clinical marker, noting that it is fast to compute and can use pairwise connectivity metrics beyond cross-correlation.They describe it as suitable for real-time monitoring systems.Source: Outlook · p. 12
  • Future evolutionary models should account for driving forces, network growth, spatial cost–efficiency trade-offs, and axonal growth patterns.These are proposed ingredients for explaining how brain-network topology developed.Source: Outlook · p. 12

A. Connectivity datasets

  • The structural datasets span C. elegans, cat, macaque, and human connectomes; the human functional dataset came from 16 healthy participants and used correlations across 116 regions.For human structural connectivity, the reported analysis uses cortical subnetworks; the averaging procedure prunes outlier links across 21 participants.Source: A. Connectivity datasets · pp. 12–14
  • The human structural networks were derived from probabilistic tractography and averaged across 21 participants; the paper reports cortical subnetworks and qualitatively similar full-brain results.The full-brain network differed in that the cerebellum and vermis formed a densely interconnected community that synchronized easily.Source: A. Connectivity datasets · p. 13

B. Community detection

The methods identify communities in the connectomes and compare network models and surrogates. The proposed hierarchical model centralizes cross-module links through preferential attachment rather than connecting modules randomly.

B. Community detection

  • The cat network’s four communities follow functional subdivisions; the C. elegans, macaque, and human partitions were detected with a staged community-detection procedure.The procedure uses Newman–Girvan identification, Tabu Search, and a final reposition optimization.Source: B. Community detection · p. 14
  • The detected partitions comprise four C. elegans modules, three macaque modules, and three modules in the average human cortical connectome.Their reported modularity values are 0.417, 0.402, and 0.33, respectively.Source: B. Community detection · p. 14

C. Surrogates and synthetic network models

  • The surrogate and benchmark methods include G(N, L) random graphs, rank-weighted scale-free networks, degree-preserving rewired networks, and nested hierarchical networks with random inter-module links.The scale-free construction assigns node weights by rank and links randomly chosen node pairs until the target link count is reached.Source: C. Surrogates and synthetic network… · p. 14
  • The proposed nested hierarchical model replaces random inter-module connections with preferential attachment, centralizing links through hubs while retaining the benchmark model’s inter-module mean degrees.The construction seeds 16 random graphs of 16 nodes each and sets mean degrees of 13, 6, and 5 across the described levels.Source: C. Surrogates and synthetic network… · pp. 14–15
  • Modularity-preserving random graphs retain the original partition’s link counts within and between each pair of communities.The construction preserves modularity for that partition while randomizing the connections.Source: C. Surrogates and synthetic network… · p. 15

D. Mapping functional connectivity from anatomical connectomes

The paper maps anatomical connectivity to functional connectivity to isolate topology’s influence on collective dynamics. Its analytical mapping uses nonlinear decay to make longer paths less influential while retaining a tractable estimate.

D. Mapping functional connectivity from anatomical connectomes

  • The exponential mapping estimates functional connectivity analytically from structural connectivity, using connection strength as its only free parameter and avoiding time-consuming simulations.The mapping is intended to focus on topology while limiting other influences on collective dynamics.Source: D. Mapping functional connectivity… · p. 15
  • The Gaussian diffusion baseline is limited to weak coupling; normalizing coupling by the largest eigenvalue gives convergence for g ∈ [0, 1) when α = 1.For directed networks, the normalization uses the largest norm of the adjacency matrix’s complex eigenvalues.Source: D. Mapping functional connectivity… · p. 15
  • Unlike the linear Gaussian propagator, which weights paths of all lengths equally and may diverge, the exponential mapping converges and favors shorter paths.The paper reports consistency with simulations using Kuramoto oscillators and neural-mass models.Source: D. Mapping functional connectivity… · p. 16
  • The exponential propagator models local perturbations diffusing through the network, with longer paths attenuated faster and correlations strengthening as coupling g grows.At weak coupling, correlations remain local; stronger coupling lets perturbations propagate farther.Source: D. Mapping functional connectivity… · p. 16

Supplementary material for: “Functional complexity emerging from anatomical constraints in the brain: the

The supplementary material expands on the functional-complexity measure, its comparisons, and the network analyses. It also compares the exponential mapping with simulations and presents rich-club analyses.

Supplementary material

  • The supplement illustrates spatial functional complexity with correlation matrices for neural connectomes and synthetic network models.The matrices show networks along the transition to global synchrony.Source: Supplementary material for:… · p. 19
  • It compares four candidate measures and reports that the integral-based measure is superior to the alternatives.The supplement also revisits Tononi, Sporns and Edelman’s neural-complexity measure, emphasizing its limitations and comparing it with the proposed measure.Source: Supplementary material for:… · p. 19
  • The supplement compares the exponential mapping with simulations of generic dynamics and includes rich-club analyses for the studied networks and hierarchical models.The rich-club analyses include the new modular hierarchical model and the Ravasz-Barabási model.Source: Supplementary material for:… · p. 19

I. EVOLUTION OF CORRELATION MATRICES WITH COUPLING

Functional complexity varies with coupling as pairwise correlations shift from near independence to synchrony. Broad correlation distributions arise at intermediate coupling, where partial coalitions form.

I. Evolution of correlation matrices with coupling

  • At intermediate coupling, partial coalitions broaden the distribution of pairwise correlations, marking the emergence of complex behavior.At weak coupling the distribution is narrow near zero; at strong coupling it narrows again near one.Source: §I · p. 19
  • The Ravasz-Barabási model retains a narrow correlation peak across coupling, indicating a lack of complex dynamics.Source: §I · p. 19
  • The neural-connectome matrices show correlation values on a common scale, with blue for rij = 0 and red for rij = 1.The matrices are shown as coupling strength g increases.Source: Figure S1 · p. 20
  • The synthetic-model matrices use the same correlation color scale, but show one network sample rather than ensemble averages.They are displayed as coupling strength g increases.Source: Figure S2 · p. 21

II. MEASURES OF FUNCTIONAL COMPLEXITY

The paper compares four ways to quantify the breadth of pairwise-correlation distributions. The integral-based measure best distinguishes the cat network from random graphs and is robust to bin-count changes.

II. Measures of functional complexity

  • The four candidate measures quantify distribution breadth using entropy, variance, Euclidean distance from a uniform distribution, or the area between distribution curves.The final two compare p with the uniform distribution; the integral measure uses the area between their curves.Source: §II · p. 22
  • The Euclidean-distance option treats the estimated distribution and the uniform distribution as vectors in an m-dimensional space.The passage introduces this as the total Euclidean distance between the two curves.Source: §II · p. 22
  • The integral measure best discriminates the cat network from equivalent random graphs, with peak-complexity ratio rint = 2.048.The other ratios are rent = 1.296, rvar = 1.046, and reuc = 1.190; entropy, variance, and Euclidean-distance measures tend to overestimate complexity, especially for random graphs.Source: §II · p. 22, Figure S3 · p. 23
  • Only the integral-based measure remains robust when the number of bins used to estimate p(rij) changes.The other three measures depend on the bin count.Source: §II · p. 22, Figure S4 · p. 23

III. LIMITATIONS OF THE NEURAL COMPLEXITY MEASURE

Neural complexity was designed to capture a balance between segregation and integration, but its behavior at global synchrony conflicts with that aim. Its all-bipartitions calculation also limits practical use to small networks.

Measure definition

  • Neural complexity sums average mutual information across bipartitions of sizes k = 1 through N/2 to quantify the coexistence of local and global coherent behavior.The intended maximum corresponds to a balance between segregation and integration.Source: §III · p. 24

Bipartitions

  • Each bipartition divides the network into complementary subsets of sizes k and N − k, and their mutual information can be computed through integration.Integration generalizes mutual information to more than two variables.Source: §III · p. 24

Computational cost

  • Computing neural complexity requires mutual information for every possible bipartition, making it feasible only for very small networks unless a random sample is used.The number of bipartitions grows with network size; sampling can partly reduce the computational burden.Source: §III · p. 24

Measure behavior

  • At global synchrony, neural complexity diverges to infinity, although the system has no segregation and therefore lacks the intended balance of segregation and integration.In contrast, functional complexity vanishes again at strong coupling in the comparison on the same correlation matrix.Source: Figure S5 · p. 24, §III · p. 25

IV. COMPARISON OF THE EXPONENTIAL MAPPING WITH DYNAMICAL MODELS

The authors compare an exponential correlation mapping with several dynamical models to assess whether it plausibly approximates collective dynamics. Across the cat network simulations, functional complexity has a similar intermediate-coupling peak across the four models.

Mapping rationale

  • The exponential mapping is presented as an analytically estimated approximation of time-averaged correlations that accounts for nonlinear signal decay over longer paths.Its single parameter, coupling strength, allows direct comparison between structural and functional connectivity.Source: §IV · p. 25

Model comparison

  • The comparison uses cat corticocortical connectivity simulated with Gaussian-noise diffusion, Kuramoto oscillators, and neural-mass models alongside the exponential mapping.The Kuramoto and neural-mass models represent oscillatory and neural activity, respectively.Source: §IV · p. 25
  • Functional complexity vanishes at the coupling extremes and peaks between them across four cat-network dynamical models, with similar peak values of 0.5–0.6.The neural-mass model does not fully reach global synchrony because of external noise.Source: §IV · p. 26

Interpretation

  • The similar complexity profiles across the four dynamical models support the plausibility of the exponential mapping as a proxy for collective dynamics in generic oscillators.The authors also note that the linear Gaussian model’s interesting regime occurs near the coupling at which the system diverges.Source: §IV · pp. 26–27

V. RICH-CLUB OF NEURAL AND SYNTHETIC NETWORKS

The section uses k-density to characterize rich-clubs in real and synthetic networks. Real networks show increasing density among high-degree nodes, while random and modularity-preserving null networks rise more slowly; the centralised hierarchical modular model forms a rich-club unlike the Ravasz–Barabási model.

k-density measure

  • k-density measures link density among nodes above a degree threshold, and the curve is evaluated across thresholds to assess whether density rises, stays stable, or falls.A falling curve rules out a rich-club; a rising curve may indicate one.Source: §V · p. 27

Degree measure

  • For directed networks, node degree is approximated by the average of its input and output degrees, justified by reciprocal connections and correlated input and output degrees.The authors note that separate rankings by input or output degree are possible for more detailed applications.Source: §V · p. 27

Real and null networks

  • Real-network k-density curves rise to large values, while random and modularity-preserving nulls increase more slowly and reach only slightly above the network’s initial density.Degree-preserving rewired networks track real networks closely, though real networks have the largest densities at the highest degrees.Source: §V · p. 27

Synthetic networks

  • The centralised hierarchical modular model forms a rich-club, whereas the Ravasz–Barabási model does not despite its scale-free-like degree distribution.The centralised model’s k-density curve is shown for the parameters used in the main-text results.Source: §V · p. 27

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